Chapter map

1. 1. Paired and unpaired design
2. 2. Feature harmonization
3. 3. RNA encoder posterior
4. 4. ATAC encoder posterior
5. 5. Negative binomial RNA decoder
6. 6. Bernoulli ATAC decoder
7. 7. Joint latent average
8. 8. Symmetric Jeffrey divergence
9. 9. Evidence lower bound by availability
10. 10. Adversarial domain alignment
11. 11. KL warm-up and early stopping
12. 12. Missing-modality decoding
13. 13. Uncertainty through sampling
14. 14. Worked two-Gaussian alignment
15. 15. Tri-modal extension
16. 16. Governance and learning

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1. Paired and unpaired design

This step deserves a careful pause. The input can include cells measured in multiple modalities and cells measured in only one.

A concrete situation

A study combines multiome, RNA-only and ATAC-only cells.

What the mathematics does not license

Study and modality can be confounded. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Tabulate every combination. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Feature harmonization

The useful question is not only how to calculate, but what the calculation assumes. RNA genes and ATAC peaks receive stable vocabularies and preprocessing before modelling.

A concrete situation

Three studies share a declared gene and peak set.

What the mathematics does not license

Intersecting features can discard informative regions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report retained proportions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. RNA encoder posterior

A small example makes the mechanism visible. An RNA network estimates a distribution q(zR) from gene counts and batch information.

A concrete situation

A cell receives a latent mean and variance from RNA.

What the mathematics does not license

Depth can dominate the encoder. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect depth correlations. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. ATAC encoder posterior

This is where modelling choices begin to affect evidence. An accessibility network estimates q(zA) from binary or count-derived peak observations.

A concrete situation

Open regions produce a second latent distribution.

What the mathematics does not license

Sparse peaks create broad uncertainty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report uncertainty by depth. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Negative binomial RNA decoder

The central mathematical idea is concrete. RNA counts use a cell-depth scaling factor, normalized gene frequencies and gene dispersion.

A concrete situation

A predicted mean of five allows overdispersed counts.

What the mathematics does not license

A poor fit biases imputation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check residuals by gene. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Bernoulli ATAC decoder

Here the geometry becomes operational. Each peak receives an accessibility probability and the observed binary state contributes log Bernoulli likelihood.

A concrete situation

Peak p is open with probability 0.72.

What the mathematics does not license

Binarization discards fragment count information. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare count-aware alternatives. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Joint latent average

This step deserves a careful pause. For paired cells, modality-specific latent parameters are combined into an average representation.

A concrete situation

RNA mean 1.0 and ATAC mean 1.4 contribute to 1.2 in one dimension.

What the mathematics does not license

Equal averaging may not reflect reliability. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare weighted alternatives. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Symmetric Jeffrey divergence

The useful question is not only how to calculate, but what the calculation assumes. KL(qA||qR) plus KL(qR||qA) penalizes mismatched paired-cell posteriors in both directions.

A concrete situation

Two close Gaussians receive a small penalty.

What the mathematics does not license

Excess pressure can erase modality-specific biology. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Sweep the weight. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Evidence lower bound by availability

A small example makes the mechanism visible. Paired cells contribute both modality likelihoods; unpaired cells contribute only the observed likelihood plus regularization.

A concrete situation

An RNA-only cell has no ATAC reconstruction target.

What the mathematics does not license

Missingness may not be random. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Model the collection process. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Adversarial domain alignment

This is where modelling choices begin to affect evidence. A classifier tries to identify modality or batch from the latent space while the encoder learns to reduce separability.

A concrete situation

RNA-only and ATAC-only cells mix more closely.

What the mathematics does not license

Perfect mixing can destroy biology. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Track conservation metrics. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. KL warm-up and early stopping

The central mathematical idea is concrete. Regularization grows during initial epochs and validation reconstruction controls stopping.

A concrete situation

KL weight rises over the first fifty epochs in the reported default.

What the mathematics does not license

Defaults are not universal constants. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Retune on a held-out design. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Missing-modality decoding

Here the geometry becomes operational. A cell state inferred from RNA is passed through the ATAC decoder, or vice versa, to form a posterior prediction.

A concrete situation

RNA-only cells receive peak probabilities.

What the mathematics does not license

Prediction can regress toward common states. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report rare-state error. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Uncertainty through sampling

This step deserves a careful pause. Repeated latent and decoder samples form intervals or variability estimates for missing values.

A concrete situation

Peak probability varies from 0.3 to 0.6 across posterior samples.

What the mathematics does not license

Intervals can be miscalibrated. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use held-out coverage tests. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Worked two-Gaussian alignment

The useful question is not only how to calculate, but what the calculation assumes. Calculate KL in both directions for one-dimensional normal posteriors, add them, then average the means for a paired cell.

A concrete situation

Different variances make the two KL terms unequal.

What the mathematics does not license

One dimension hides covariance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Extend to diagonal vectors. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Tri-modal extension

A small example makes the mechanism visible. A protein encoder and decoder add a third posterior, with pairwise alignment across available modalities.

A concrete situation

RNA, ATAC and protein share a cell state.

What the mathematics does not license

Pairwise agreement may be inconsistent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect every modality pair. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

This is where modelling choices begin to affect evidence. Version pairing, features, likelihoods, encoders, alignment and imputation. This joins distributions, divergence and missing-data reasoning.

A concrete situation

Students can begin with Bernoulli and KL.

What the mathematics does not license

An imputation is a conditional estimate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a MultiVI evidence card. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

A MultiVI audit starts by naming each modality and its observation law. RNA counts use a negative binomial likelihood; chromatin-accessibility indicators use a Bernoulli likelihood; additional modalities receive their own encoders and decoders. For paired cells, modality-specific latent distributions are averaged while a symmetric KL penalty encourages them to describe the same cell state.

Record paired and unpaired cells, genes, peaks and proteins, feature harmonization, library-size variables, modality encoders, shared latent dimension, decoders, symmetric Jeffrey divergence, batch adversary, KL warm-up, validation split and early stopping. A missing modality is inferred from the learned joint distribution, so its uncertainty must travel with the prediction.

Validate by hiding genuinely observed modalities, holding out studies and technologies, comparing paired-only and unpaired regimes, measuring modality mixing and biological preservation, and checking uncertainty calibration. Imputed accessibility or expression can support hypotheses and prioritization, but it is not laboratory observation and should not silently replace a missing assay.

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Frequently asked questions

What is MultiVI?

MultiVI is a deep generative model for integrating multimodal and single-modality single-cell data. The 2023 paper focuses on RNA and chromatin accessibility and demonstrates extension to protein.

How does MultiVI handle paired cells?

Each available modality has its own encoder. For paired cells, modality-specific posterior distributions are encouraged to agree, and their representations contribute to a joint latent state.

How does it handle an unpaired cell?

If only one modality is observed, the cell state is drawn from that modality’s posterior. The matching decoder can then estimate a missing modality under the learned joint model.

Why use different likelihoods?

RNA count variability is modelled with a negative binomial distribution, whereas chromatin accessibility is treated as a Bernoulli outcome. The difference reflects the mathematical form of the measurements.

What is symmetric KL alignment?

It adds KL(qA||qR) and KL(qR||qA), penalizing disagreement in both directions between accessibility and RNA posterior distributions for paired cells.

Is imputed chromatin accessibility observed data?

No. It is a posterior prediction conditional on the training data, model and observed modality. The estimate needs uncertainty and validation against genuinely measured held-out assays.

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Primary source and fact-check note

Tal Ashuach, Mariano I. Gabitto, Rohan V. Koodli et al., “MultiVI: deep generative model for the integration of multimodal data,” Nature Methods 20, 1222–1231 (2023), DOI 10.1038/s41592-023-01909-9. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• CellPLM multimodal latent variables
• scPRINT read-depth-aware representations
• Nicheformer spatial context transfer
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Pairing audit

Report how many cells have every combination of RNA, ATAC and protein measurements.

Feature audit

Version genes, peaks, genome build, filters and shared-feature construction.

Likelihood check

Test negative binomial RNA and Bernoulli accessibility with held-out observations.

Alignment ledger

Report symmetric KL, prior KL, reconstruction and domain-adaptation terms separately.

Masked-modality benchmark

Hide actually measured modalities and score prediction, calibration and rare states.

Missingness challenge

Test whether study, cell type or quality predicts which modality is absent.

Study holdout

Reserve complete technologies and cohorts, not random cells from the same sample.

Final claim

Name pairing, likelihoods, alignment and decoder. An inferred modality is not an assay result.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Paired and unpaired design

Start with this article's mechanism: The input can include cells measured in multiple modalities and cells measured in only one. Work through the concrete case: A study combines multiome, RNA-only and ATAC-only cells. Make the boundary visible: Study and modality can be confounded. Complete the evidence step: Tabulate every combination. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 2: Feature harmonization

Start with this article's mechanism: RNA genes and ATAC peaks receive stable vocabularies and preprocessing before modelling. Work through the concrete case: Three studies share a declared gene and peak set. Make the boundary visible: Intersecting features can discard informative regions. Complete the evidence step: Report retained proportions. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 3: RNA encoder posterior

Start with this article's mechanism: An RNA network estimates a distribution q(zR) from gene counts and batch information. Work through the concrete case: A cell receives a latent mean and variance from RNA. Make the boundary visible: Depth can dominate the encoder. Complete the evidence step: Inspect depth correlations. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 4: ATAC encoder posterior

Start with this article's mechanism: An accessibility network estimates q(zA) from binary or count-derived peak observations. Work through the concrete case: Open regions produce a second latent distribution. Make the boundary visible: Sparse peaks create broad uncertainty. Complete the evidence step: Report uncertainty by depth. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 5: Negative binomial RNA decoder

Start with this article's mechanism: RNA counts use a cell-depth scaling factor, normalized gene frequencies and gene dispersion. Work through the concrete case: A predicted mean of five allows overdispersed counts. Make the boundary visible: A poor fit biases imputation. Complete the evidence step: Check residuals by gene. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 6: Bernoulli ATAC decoder

Start with this article's mechanism: Each peak receives an accessibility probability and the observed binary state contributes log Bernoulli likelihood. Work through the concrete case: Peak p is open with probability 0.72. Make the boundary visible: Binarization discards fragment count information. Complete the evidence step: Compare count-aware alternatives. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 7: Joint latent average

Start with this article's mechanism: For paired cells, modality-specific latent parameters are combined into an average representation. Work through the concrete case: RNA mean 1.0 and ATAC mean 1.4 contribute to 1.2 in one dimension. Make the boundary visible: Equal averaging may not reflect reliability. Complete the evidence step: Compare weighted alternatives. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 8: Symmetric Jeffrey divergence

Start with this article's mechanism: KL(qA||qR) plus KL(qR||qA) penalizes mismatched paired-cell posteriors in both directions. Work through the concrete case: Two close Gaussians receive a small penalty. Make the boundary visible: Excess pressure can erase modality-specific biology. Complete the evidence step: Sweep the weight. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 9: Evidence lower bound by availability

Start with this article's mechanism: Paired cells contribute both modality likelihoods; unpaired cells contribute only the observed likelihood plus regularization. Work through the concrete case: An RNA-only cell has no ATAC reconstruction target. Make the boundary visible: Missingness may not be random. Complete the evidence step: Model the collection process. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 10: Adversarial domain alignment

Start with this article's mechanism: A classifier tries to identify modality or batch from the latent space while the encoder learns to reduce separability. Work through the concrete case: RNA-only and ATAC-only cells mix more closely. Make the boundary visible: Perfect mixing can destroy biology. Complete the evidence step: Track conservation metrics. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 11: KL warm-up and early stopping

Start with this article's mechanism: Regularization grows during initial epochs and validation reconstruction controls stopping. Work through the concrete case: KL weight rises over the first fifty epochs in the reported default. Make the boundary visible: Defaults are not universal constants. Complete the evidence step: Retune on a held-out design. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 12: Missing-modality decoding

Start with this article's mechanism: A cell state inferred from RNA is passed through the ATAC decoder, or vice versa, to form a posterior prediction. Work through the concrete case: RNA-only cells receive peak probabilities. Make the boundary visible: Prediction can regress toward common states. Complete the evidence step: Report rare-state error. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 13: Uncertainty through sampling

Start with this article's mechanism: Repeated latent and decoder samples form intervals or variability estimates for missing values. Work through the concrete case: Peak probability varies from 0.3 to 0.6 across posterior samples. Make the boundary visible: Intervals can be miscalibrated. Complete the evidence step: Use held-out coverage tests. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 14: Worked two-Gaussian alignment

Start with this article's mechanism: Calculate KL in both directions for one-dimensional normal posteriors, add them, then average the means for a paired cell. Work through the concrete case: Different variances make the two KL terms unequal. Make the boundary visible: One dimension hides covariance. Complete the evidence step: Extend to diagonal vectors. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 15: Tri-modal extension

Start with this article's mechanism: A protein encoder and decoder add a third posterior, with pairwise alignment across available modalities. Work through the concrete case: RNA, ATAC and protein share a cell state. Make the boundary visible: Pairwise agreement may be inconsistent. Complete the evidence step: Inspect every modality pair. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 16: Governance and learning

Start with this article's mechanism: Version pairing, features, likelihoods, encoders, alignment and imputation. This joins distributions, divergence and missing-data reasoning. Work through the concrete case: Students can begin with Bernoulli and KL. Make the boundary visible: An imputation is a conditional estimate. Complete the evidence step: Write a MultiVI evidence card. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.